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Heat Transfer Handbook part 23

Heat Transfer Handbook part 23. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | EXTENDED SURFACES 211 Figure 3.25 Efficiencies of convecting spines. 3.6.4 Longitudinal Radiating Fins Unlike convecting fins for which exact analytical solutions abound few such solutions are available for radiating fins. Consider the longitudinal fin of rectangular profile shown in Fig. 3.19a and let the fin radiate to free space at 0 K. The differential equation governing the temperature in the fin is d1T 2oe 4 --- ---T4 dx2 k 3.212 with the boundary conditions T x 0 Tb and dT dx 0 x b 3.213 where e is the emissivity of the tin surfaee ami a is the Stefan-Boltzmann constant o 5.667 x IO-8 W m2 K4 . The solution for the temperature distribution rate of heat transfer and fin efficiency are B 0.3 0.5 - Bu 0.3 0.5 b 20 T- k 8 3.214 qf 2k8L 1 2 Tb5 - Tz5 1 2 3.215 5k0 212 CONDUCTION HEAT TRANSFER 2ML 5k 2 T - C f 2 n -------------------------- 3.216 2aebLTb4 where B and Bu are complete and incomplete beta functions discussed in Section 3.3.3 u Tt T 5 and Tt is the unknown tip temperature. Because Tt is not known the solution involves a trial-and-error procedure. Sen and Trinh 1986 reported the solution of eqs. 3.212 and 3.213 when the surface heat dissipation is proportional to Tm rather than T4. Their solution appears in terms of hypergeometric functions which bear a relationship to Hie incomplete he tit function. Kraus et al. 2001 provide an extensive collection of graphs to evaluate the performance or radiati rig fins of difi etent fireflies. 3.6.5 Longitudinal Convecting-Radiating Fins A flnite-difference approach was taken by Nguyen and Aziz 1992 to evaluate the peaCormence oP i glgitflninal nns Fig. 39t9o o1 I elngngl.llar irape taelal. ihangl.ilar. and concave parabolic proflles when the fln surface loses heat by simultaneous convection and radiation. or each proflle the performance depends on flve parameters 2b b hlb 2k Tx Tb Ts Tb and 2b2 rnTb klb where Ts is theeffective sink temperature for radiation. A sample result for the fln efficiency is provided in .